Prandtl-Meyer reflection configurations, transonic shocks, and free boundary problems / Myoungjean Bae, Gui-Qiang G. Chen, Mikhail Feldman.

Format:

Book

Author/Creator:

Bae, Myoungjean, author.

Chen, Gui-Qiang, 1963- author.

Feldman, Mikhail, 1960- author.

Series:

Memoirs of the American Mathematical Society ; v. 1507.

Memoirs of the American Mathematical Society ; v. 1507

Language:

English

Subjects (All):

Differential equations, Partial.

Mathematical physics.

Fluid dynamics.

Physical Description:

vii, 237 pages : illustrations ; 26 cm.

Place of Publication:

Providence, Rhode Island : American Mathematical Society, 2024.

Summary:

We are concerned with the Prandtl-Meyer reflection configurations of unsteady global solutions for supersonic flow impinging upon a symmetric solid wedge. Prandtl (1936) first employed the shock polar analysis to show that there are two possible steady configurations: the steady weak and strong shock solutions, when a steady supersonic flow impinges upon the wedge whose angle is less than the detachment angle, and then conjectured that the steady weak shock solution is physically admissible. The fundamental issue of whether one or both of the steady weak/strong shocks are physically admissible has been vigorously debated over the past eight decades and has not yet been settled definitively. On the other hand, the Prandtl-Meyer reflection configurations are core configurations in the structure of global entropy solutions of the 2-D Riemann problem, while the Riemann solutions themselves are local building blocks and determine local structures, global attractors, and large-time asymptotic states of general entropy solutions. In this sense, we have to understand the reflection configurations to understand fully the global entropy solutions of 2-D hyperbolic systems of conservation laws, including the admissibility issue for the entropy solutions. In this monograph, we address this longstanding open issue and present our analysis to establish the stability theorem for the steady weak shock solutions as the long-time asymptotics of the Prandtl-Meyer reflection configurations for unsteady potential flow for all the physical parameters up to the detachment angle. To achieve these, we first reformulate the problem as a free boundary problem involving transonic shocks and then obtain appropriate monotonicity properties and uniform a priori estimates for admissible solutions, which allow us to employ the Leray-Schauder degree argument to complete the theory.

Notes:

"September 2024 ; vol. 301 number 1507 (first of 7 numbers)".

Includes bibliographical references (pages 235-237).

ISBN:

1470462702

9781470462703

OCLC:

1467670769